Please help me hurry I need to finish the table I’ll mark brainly

f · dr c = 158/3 for the given f = x2 i y2 j and c is the line from point (5, 4) to point (7, 6).

To find f · dr c for the given f and c, we must first parameterize the line segment c. We can do this by letting x = 5 + t(2) and y = 4 + t(2), where 0 ≤ t ≤ 1. This gives us the vector equation r(t) = 5i + 4j + 2ti + 2tj.

Next, we need to calculate the r(t) differential, which is dr = 2i dt + 2j dt. We can then rewrite this as dr = (2i + 2j) dt.

Now we can calculate f · dr c by substituting our parameterizations into the dot product formula:

f · dr c = ∫f · dr = ∫(x2 i + y2 j) · (2i + 2j) dt

= ∫(2x2 + 2y2) dt

= ∫(2[(5 + 2t)2] + 2[(4 + 2t)2]) dt

= ∫(50 + 40t + 8t2) dt

= 50t + 20t2 + (8/3)t3 + C

evaluated from t = 0 to t = 1.

Plugging in our values, we get:

f · dr c = (50 + 20 + (8/3)) - (0 + 0 + 0) = 158/3

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According to Newton's Law of Cooling,  it takes 90 minutes for the chicken to reach 20°C.

According to Newton's Law of Cooling, the rate at which an object's temperature changes is proportional to the difference between its temperature and the ambient temperature. The formula for Newton's Law of Cooling is:

ΔT/Δt = k(T - Ta)

Where ΔT is the change in temperature, Δt is the change in time, k is a constant, T is the object's temperature, and Ta is the ambient temperature.

From the given information, we have:

ΔT1 = 10C - 0C = 10°C
Δt1 = 45 minutes
Ta = 23°C

Now, we want to find the time it takes for the chicken to reach 20°C:

ΔT2 = 20C - 0C = 20°C

Using the formula and the fact that k and Ta are constants, we can set up the following proportion:

(ΔT1/Δt1) / (ΔT2/Δt2) = 1

Solving for Δt2:

(10/45) / (20/Δt2) = 1

Cross-multiplying and solving for Δt2, we get:

Δt2 = 90 minutes

So, it takes 90 minutes for the chicken to reach 20°C.

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Answer:

B

Mark brainliest and help feed my children

(also thank and rate 5 stars)

hope this helped

Let n ≥ 1, x be a real number, and x≥ −1.

By mathematical induction, we have shown that (1 + x)n ≥ 1 + nx for all n ≥ 1 and x ≥ -1.

To prove that (1 + x)n ≥ 1 + nx for all n ≥ 1 and x ≥ -1 using mathematical induction,

we need to first establish a base case and then show that if the statement holds for n = k, it also holds for n = k + 1.

Base case: When n = 1, we have (1 + x)^1 = 1 + x and 1 + 1x = 1 + x. Therefore, the statement is true for n = 1.

Inductive step:

Assume that (1 + x)k ≥ 1 + kx for some arbitrary positive integer k. We want to show that (1 + x)k+1 ≥ 1 + (k + 1)x.

Starting with the left-hand side of the inequality:

(1 + x)k+1 = (1 + x)k (1 + x)

By the inductive hypothesis, we know that (1 + x)k ≥ 1 + kx, so we can substitute that in:

(1 + x)k+1 ≥ (1 + kx)(1 + x)

Expanding the right-hand side:

(1 + kx)(1 + x) = 1 + kx + x + kx^2 = 1 + (k + 1)x + kx^2

So we have:

(1 + x)k+1 ≥ 1 + (k + 1)x + kx^2

Now, since x ≥ -1, we know that kx^2 ≥ -k. Adding this to both sides of the inequality, we get:

(1 + x)k+1 + k ≥ 1 + (k + 1)x

Finally, since k is a positive integer, we know that (1 + x)k+1 + k ≥ 1 + (k + 1)x, which completes the inductive step.

Therefore, by mathematical induction, we have shown that (1 + x)n ≥ 1 + nx for all n ≥ 1 and x ≥ -1.

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Every minute, the number of bacteria decays by a factor of 16^(-60).

An exponential function has the definition presented as follows:

y = ab^x.

In which the parameters are given as follows:

The decay factor k of the exponential function is obtained as follows:

b = 1 - k

k = 1 - b.

The parameter b for the function in this problem is given as follows:

b = 15/16.

Hence the decay factor each second is obtained as follows:

k = 1 - 15/16

k = 16/16 - 15/16

k = 1/16.

Then the decay factor each minute is given as follows:

k = (1/16)^60

k = 16^(-60).

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The total cost for transporting the cars will be (2*7) + (3*11) + (5*8) + (7*5) = 94 car miles.
Therefore, the rent-a-car company should follow this plan to meet the supply requirements at a minimum cost.

To plan the day's activity such that supply requirements are met at a minimum cost, we can use the transportation problem method. We will create a matrix with rows representing the branches and columns representing the destinations. The cells will represent the number of cars transported from each branch to each destination.

We start by filling the cells with the lowest transportation cost. For example, from branch 1 to destination A, the cost is 7, which is the lowest cost among all the other options. We will continue filling the cells with the lowest costs until we have met the supply requirements for each destination.

Here is the completed matrix:

Destination A B C D Supply

Branch 1 2 0 0 0 2

Branch 2 0 3 0 0 3

Branch 3 0 0 5 7 12

Demand 2 3 5 7

To interpret the matrix, we can see that branch 1 will supply 2 cars to destination A and branch 2 will supply 3 cars to destination B. Branch 3 will supply 5 cars to destination C and 7 cars to destination D. The total cost for transporting the cars will be (2*7) + (3*11) + (5*8) + (7*5) = 94 car-miles.

Therefore, the rent-a-car company should follow this plan to meet the supply requirements at a minimum cost

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Answer:

-31 c + 13

Step-by-step explanation:

-13c+8-18c+5

combine like terms

-18c and -13c combined is -31c

8 + 5 = 13

-31 c + 13

Answer

-31c+13 is your answer (see explanation below!)

Step-by-step explanation:

1) Add the numbers:

[tex]-13c + 8 - 18c + 5\\-13c + 13 -18c\\[/tex]

2) Combine like terms:

[tex]-13c+13-18c\\-31c+13\\[/tex]

[tex]A: -31c+13[/tex]

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1. The probability of gaining no more than 2 clients is 0.28. 2. For atleast 4 new clients: 0.37. 3. The expected value is 3.17.

The study of arbitrary events or experiments falls under the purview of probability, a subfield of mathematics. It is used to determine how likely an event is to occur, with a range of 0 (impossible) to 1. (certain). In a variety of domains, including economics, engineering, physics, and social sciences, probability can be used to assess and forecast events. It entails applying formulae, equations, and statistical analysis to calculate the probabilities of a specific event occurring under specific circumstances or presumptions. Decision-making, risk management, and many other aspects of daily life all depend on the concept of probability.

1. The probability of gaining no more than 2 clients is given as:

P(X ≤ 2) = P(X = 0) + P(X = 1) + P(X = 2)

Substituting the value of probabilities from the table we have;

P(X ≤ 2) = 0.03 + 0.10 + 0.15 = 0.28

2. For atleast 4 new clients we have:

P(X ≥ 4) = P(X = 4) + P(X = 5) + P(X = 6) = 0.20 + 0.10 + 0.07 = 0.37

3. The expected value is given as:

E(X) = 0(0.03) + 1(0.10) + 2(0.15) + 3(0.35) + 4(0.20) + 5(0.10) + 6(0.07) = 3.17

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The complete question is:

New clients, X : 0 1 2 3 4 5 6

P(X): 0.03 0.10 0.15 0.35 0.20 0.10 0.07

1. The probability of gaining no more than 2 clients is 0.28. 2. For atleast 4 new clients: 0.37. 3. The expected value is 3.17.

The study of arbitrary events or experiments falls under the purview of probability, a subfield of mathematics. It is used to determine how likely an event is to occur, with a range of 0 (impossible) to 1. (certain). In a variety of domains, including economics, engineering, physics, and social sciences, probability can be used to assess and forecast events. It entails applying formulae, equations, and statistical analysis to calculate the probabilities of a specific event occurring under specific circumstances or presumptions. Decision-making, risk management, and many other aspects of daily life all depend on the concept of probability.

1. The probability of gaining no more than 2 clients is given as:

P(X ≤ 2) = P(X = 0) + P(X = 1) + P(X = 2)

Substituting the value of probabilities from the table we have;

P(X ≤ 2) = 0.03 + 0.10 + 0.15 = 0.28

2. For atleast 4 new clients we have:

P(X ≥ 4) = P(X = 4) + P(X = 5) + P(X = 6) = 0.20 + 0.10 + 0.07 = 0.37

3. The expected value is given as:

E(X) = 0(0.03) + 1(0.10) + 2(0.15) + 3(0.35) + 4(0.20) + 5(0.10) + 6(0.07) = 3.17

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The complete question is:

New clients, X : 0 1 2 3 4 5 6

P(X): 0.03 0.10 0.15 0.35 0.20 0.10 0.07

In a circle with center O, chord KL is perpendicular to diameter GH. If KP=PL=18 and GH=36, what is the measure of arc GM?

Based on the mentioned informations and provided valus, the measure of arc of the circle GM is calculated out to be 18π.

Since KL is perpendicular to GH and GH is a diameter, KL is a chord that bisects the circle into two equal halves. Therefore, the arc GM is half the measure of the circle.

The measure of the circle can be found using the diameter GH, which is equal to 36. The formula for the circumference of a circle is C = πd, where d is the diameter. Therefore, the circumference of this circle is C = π(36) = 36π.

Since arc GM is half the measure of the circle, its measure can be found by dividing the circumference by 2.

arc GM = (1/2)C = (1/2)(36π) = 18π

Therefore, the measure of arc GM is 18π.

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The polynomials are

In order to find the factored form of a polynomial with x-intercepts at (-3, 0), (2, 0), and (4, 0), we must write out the equation as:

f(x) = a * (x + 3) * (x - 2) * (x - 4)

Knowing that a = 1, we simplify the equation to obtain the final form:

f(x) = (x + 3) * (x - 2) * (x - 4)

If the given curve has a bounce at the point (2,0) and a bend at (8,0), then its factored form would be:

f(x) = a * (x - 2)^2 * (x - 8)

Given that a = 1, the simplified version is written as follows:

f(x) = (x - 2)^2 * (x - 8)

Using (3, -6),

y = a * x^2 * (x + 1)

solving for a as follows:

-6 = a * 3^2 * (3 + 1)

-6 = a * 9 * 4

a = -6 / 36

f(x) = (-1/6) * x^2 * (x + 1).

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If the student who retook the test had originally scored the lowest or one of the lowest scores, then the new range might not change much, if at all.

However, if the student had originally scored somewhere in the middle or towards the higher end of the range, then the new range will likely be larger than the original range, because 78 is higher than most of the original scores.

The mean (average) of a set of numbers is calculated by adding up all the numbers and dividing the sum by the total number of numbers.

mean = S/n

Therefore, the new mean score will be:

new mean = (S + 23)/n

The range of a set of numbers is the difference between the highest and lowest numbers in the set.

Before the retest, let's say the lowest score was a, and the highest score was b. Then, the range was:

range = b - a

After the retest, the lowest score will still be a, but the highest score will be either b or 78, whichever is higher. Therefore, the new range will be:

new range = max(b, 78) - a

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Answer:

[tex]\huge\boxed{\sf 48k - 147}[/tex]

Step-by-step explanation:

= -11 + 8(6k - 17)

= -11 + 48k - 136

= 48k - 11 - 136

[tex]\rule[225]{225}{2}[/tex]

Answer:

48k - 147

Step-by-step explanation:

Now we have to,

→ Simplify the given expression.

The expression is,

→ -11 + 8(6k - 17)

Major steps we use are,

→ Rearranging the expression.

→ Combining the like terms.

Let's simplify the expression,

→ -11 + 8(6k - 17)

→ 8(6k - 17) - 11

→ 8(6k) - 8(17) - 11

→ 48k - 136 - 11

→ 48k - (136 + 11)

→ 48k - 147

Hence, the answer is 48k - 147.

The value returned by the call binary Search(values, target) is 5.

Let's perform a binary search on the given array:

The code segment provided is: int[] values = {1, 2, 3, 4, 5, 8, 8, 8}; int target = 8;

1. Initialize variables: low = 0, high = 7 (length of array - 1)
2. Calculate mid: mid = (low + high) / 2 = (0 + 7) / 2 = 3
3. Check if the target is equal to the middle element: values[3] = 4, which is not equal to 8
4. Since the target (8) is greater than the middle element (4), update low: low = mid + 1 = 3 + 1 = 4
5. Calculate mid again: mid = (low + high) / 2 = (4 + 7) / 2 = 5
6. Check if the target is equal to the middle element: values[5] = 8, which is equal to the target

As a result, the binary search function returns the index of the target, which is 5.

Therefore, the value returned by the call binary Search(values, target) is 5.

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The solution y(t) is a piecewise defined function given by: [tex]y(t) = (e^(-t/2) \times sin((t - \pi)/2))/2 + (e^(-t/2)\times sin((t - \pi)/2 + \pi))/2 for \pi \leq t \leq < 2\pi[/tex]

y(t) = 0 for t < π and t ≥ 2π

To solve the given initial value problem using Laplace transform, we apply the Laplace transform to both sides of the differential equation:

L{y''} + 2L{y'} + 2L{y} = L{g(t)}

Using the standard Laplace transform formulas for derivatives and unit step function, we get:

[tex]s^2[/tex] Y(s) - s y(0) - y'(0) + 2s Y(s) - 2y(0) + 2Y(s) = 1/(s[tex]e^(\pi)[/tex] - s e^(2π))

Substituting y(0) = 0 and y'(0) = 1, and simplifying, we get:

Y(s) = (1 - s)/([tex]s^2[/tex] + 2s + 2) [tex]\times[/tex] 1/(s [tex]e^\pi[/tex] - s [tex]e^(2\pi)[/tex])

To express y(t) as a piecewise defined function, we need to invert this Laplace transform using partial fraction decomposition and inverse Laplace transform. The roots of the denominator s^2 + 2s + 2 are complex conjugates given by:

s = -1 + i and s = -1 - i

Therefore, we can write the partial fraction decomposition as:

(1 - s)/([tex]s^2[/tex] + 2s + 2) = A/(s + 1 - i) + B/(s + 1 + i)

Multiplying both sides by the denominator and substituting s = -1 + i and s = -1 - i, we get:

A = (-1 + i)/4 and B = (-1 - i)/4

Substituting these values, we get:

Y(s) = (-1 + i)/(4(s + 1 - i)) + (-1 - i)/(4(s + 1 + i))

Taking the inverse Laplace transform of each term using the table of Laplace transforms, we get:

y(t) = ([tex]e^{(-t/2)[/tex] [tex]\times[/tex]sin((t - π)/2))/2 + ([tex]e^{(-t/2)[/tex][tex]\times[/tex]sin((t - π)/2 + π))/2 for π ≤ t < 2π

and y(t) = 0 for t < π and t ≥ 2π

Therefore, the solution y(t) is a piecewise defined function given by:

y(t) = ([tex]e^{(-t/2)[/tex] [tex]\times[/tex] sin((t - π)/2))/2 + ([tex]e^{(-t/2)[/tex][tex]\times[/tex] sin((t - π)/2 + π))/2 for π ≤ t < 2π

y(t) = 0 for t < π and t ≥ 2π

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The solution y(t) is a piecewise defined function given by: [tex]y(t) = (e^(-t/2) \times sin((t - \pi)/2))/2 + (e^(-t/2)\times sin((t - \pi)/2 + \pi))/2 for \pi \leq t \leq < 2\pi[/tex]

y(t) = 0 for t < π and t ≥ 2π

To solve the given initial value problem using Laplace transform, we apply the Laplace transform to both sides of the differential equation:

L{y''} + 2L{y'} + 2L{y} = L{g(t)}

Using the standard Laplace transform formulas for derivatives and unit step function, we get:

[tex]s^2[/tex] Y(s) - s y(0) - y'(0) + 2s Y(s) - 2y(0) + 2Y(s) = 1/(s[tex]e^(\pi)[/tex] - s e^(2π))

Substituting y(0) = 0 and y'(0) = 1, and simplifying, we get:

Y(s) = (1 - s)/([tex]s^2[/tex] + 2s + 2) [tex]\times[/tex] 1/(s [tex]e^\pi[/tex] - s [tex]e^(2\pi)[/tex])

To express y(t) as a piecewise defined function, we need to invert this Laplace transform using partial fraction decomposition and inverse Laplace transform. The roots of the denominator s^2 + 2s + 2 are complex conjugates given by:

s = -1 + i and s = -1 - i

Therefore, we can write the partial fraction decomposition as:

(1 - s)/([tex]s^2[/tex] + 2s + 2) = A/(s + 1 - i) + B/(s + 1 + i)

Multiplying both sides by the denominator and substituting s = -1 + i and s = -1 - i, we get:

A = (-1 + i)/4 and B = (-1 - i)/4

Substituting these values, we get:

Y(s) = (-1 + i)/(4(s + 1 - i)) + (-1 - i)/(4(s + 1 + i))

Taking the inverse Laplace transform of each term using the table of Laplace transforms, we get:

y(t) = ([tex]e^{(-t/2)[/tex] [tex]\times[/tex]sin((t - π)/2))/2 + ([tex]e^{(-t/2)[/tex][tex]\times[/tex]sin((t - π)/2 + π))/2 for π ≤ t < 2π

and y(t) = 0 for t < π and t ≥ 2π

Therefore, the solution y(t) is a piecewise defined function given by:

y(t) = ([tex]e^{(-t/2)[/tex] [tex]\times[/tex] sin((t - π)/2))/2 + ([tex]e^{(-t/2)[/tex][tex]\times[/tex] sin((t - π)/2 + π))/2 for π ≤ t < 2π

y(t) = 0 for t < π and t ≥ 2π

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Considering the differential equation 2x²y" + 3xy' + (2x - 1)y = 0. A series solution corresponding to the indicial root r=- 1 is y=x-'[1+372 €***), where [tex]c_k=\frac{(-2)^k}{k!(-1)*(2k-3)!}[/tex].

The given differential equation has been transformed into the indicial equation 2r²+r-1=0, which has the roots r=1/2 and r=-1. We are interested in finding a series solution corresponding to the indicial root r=-1.
To do this, we first assume a solution of the form y(x) = [tex]x^r[/tex] * Σ_[tex](n=0)^{(∞)} c_n[/tex] * [tex]x^n[/tex]. Substituting this into the given differential equation and simplifying, we get a recurrence relation for the coefficients [tex]c_n[/tex]. In this case, the recurrence relation is Cz[2(k+r)+(k+r-1)+3(k+r)-1]+202-1=0, where C is a constant and k is the index of the coefficients.
Next, we need to use the indicial root r=-1 to solve for the coefficients [tex]c_n[/tex]. Plugging in r=-1 into the assumed solution, we get y(x) = [tex]x^{-1}[/tex] * Σ[tex]_(n=0)^{(∞)} c_n[/tex] * [tex]x^n[/tex]. We can simplify this to y(x) = Σ_[tex](n=0)^{(∞)}[/tex] c_n * [tex]x^{(n-1)}[/tex]. Then, we can use the recurrence relation to solve for the coefficients.
In this case, the correct answer is [tex]c_k=\frac{(-2)^k}{k!(-1)*(2k-3)!}[/tex].

The complete question is:-

Consider the differential equation 2x²y" + 3xy' + (2x - 1)y = 0. The indicial equation is [tex]2r^2[/tex]+r-1=0. The recurrence relation is [tex]c_k{2(k+r)+(k+r-1)+3(k+r)-1]+2c_{k-1}=0[/tex].

A series solution corresponding to the indicial root r=- 1 is y=x-'[1+372 €***), where

Select the correct answer.

a. [tex]c_k=\frac{(-2)^k}{k!(-1).1.3...(2k-3)}[/tex]

b. [tex]c_k=\frac{-2^k}{k!.1.3...(2k-3)}[/tex]

c. [tex]c_k=\frac{(-2)^k}{k!(-1).1.3...(2k-1)}[/tex]

d. [tex]c_k=\frac{(-2)^k}{k!(-1)*(2k-3)!}[/tex]

e. [tex]c_k=\frac{(-2)^k}{k!(-1).1.3...(2k-5)}[/tex]

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Answer: £46.36.

Step-by-step explanation:  To decrease £61 by 24%, we first need to find 24% of £61. We can do this by multiplying £61 by 0.24: £61 * 0.24 = £14.64. Now, to decrease £61 by 24%, we subtract £14.64 from £61: £61 - £14.64 = £46.36.

So, if you decrease £61 by 24%, the result is £46.36.

Answer:

8x^(3/2)

Step-by-step explanation:

We can simplify the expression first:

2sqrt(x)4x^(-5/2)=8x^(-3/2)

Now we can rewrite this in the form kx^2:

8x^(=3/2)=8(x^(-3/2))(x^(5/2))/x^2

=8(x^2/x^3)(x^(1/2))/x^2

=8x^(-1/2)

therefore, 2sqrt(x)4x^(-5/2) is equivalent to 8x^(-1/2), which can be written in the form kx^2 as 8x^(3/2)

I hope this helps!

Answer: complementary

Step-by-step explanation:

adds to 90 degrees

Answer:

There is no solution.

Step-by-step explanation:

The graphs are parallel. They will never intersect each other.

The value of ∂z/∂x is -x/3z and the value of partial derivative ∂z/∂y is -2y/3z.

In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant. Partial derivatives are used in vector calculus and differential geometry.

The partial derivative is a way to find the slope in either the x or y direction, at the point indicated.

To find ∂z/∂x and ∂z/∂y using implicit differentiation, we first differentiate both sides of the equation with respect to x and y, respectively:

Differentiating with respect to x:
2x + 3(∂z/∂x)(2z) = 0

Simplifying, we get:
∂z/∂x = -2x/6z = -x/3z

Differentiating with respect to y:
4y + 3(∂z/∂y)(2z) = 0

Simplifying, we get:
∂z/∂y = -4y/6z = -2y/3z

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2,186 ways

If the order of objects is important and you need to select 13 objects 3 at a time, you can use permutations to find the number of ways this can be done.

Your answer: There are 2,186 ways to select 13 objects 3 at a time when order is important.

Step-by-step explanation:

1. Use the formula for permutations: P(n, r) = n! / (n - r)!, where n is the total number of objects (13) and r is the number of objects to be selected at a time (3).

2. Calculate the factorials: 13! = 6,227,020,800 and 10! = 3,628,800.

3. Divide the two factorials: 6,227,020,800 / 3,628,800 = 2,186.

So, there are 2,186 ways to select 13 objects 3 at a time when order is important.

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Values the constant c are;

a) c = 2.16.

b) c = 0.57.

c) c = -1.17.

a) We need to find the value of c such that Φ(c) = 0.9838. Using a standard normal table or a calculator, we find that the z-score corresponding to a cumulative probability of 0.9838 is approximately 2.16. Therefore, c = 2.16.

b) We need to find the value of c such that P(0 ≤ Z ≤ c) = 0.291. Using a standard normal table or a calculator, we find that the z-score corresponding to a cumulative probability of 0.291 is approximately 0.57. Therefore, c = 0.57.

c) We need to find the value of c such that P(c ≤ Z) = 0.121. Using a standard normal table or a calculator, we find that the z-score corresponding to a cumulative probability of 0.121 is approximately -1.17. Therefore, c = -1.17.

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Answer:

The equation of a circle with center (a,b) and radius r is given by:

(x - a)² + (y - b)² = r²

Comparing this with the given equation x² + y² = 196, we can see that a = 0, b = 0, and r² = 196. Therefore, the radius of the circle is:

r = sqrt(196) = 14

Hence, the radius of the circle is 14 units.

The p-value for the corresponding two-tailed test of hypothesis would be 0.026, obtained by doubling the p-value for the one-sided test.

To find the p-value for the corresponding two-tailed test of hypothesis, you would need to double the p-value for the one-sided test. This is because the p-value for a one-tailed test only considers one direction of the hypothesis, whereas the p-value for a two-tailed test considers both directions.

So, if the p-value for a one-sided test of hypothesis is p = 0.013, then the p-value for the corresponding two-tailed test of hypothesis would be

p-value = 2 × 0.013

Multiply the numbers

= 0.026

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Answer: 2 red balloons and 4 white balloons in each bunch

Step-by-step explanation:

divide 16/8 = 2 balloons in each bunch

divide 32/8 = 4 balloons in each bunch

c1 and c2 are arbitrary constants, and x is the independent variable.

We first need to find the characteristic equation:

r² + 10r + 41 = 0

Now, we need to find the roots of this quadratic equation. Using the quadratic formula:

r = (-b ± √(b² - 4ac)) / 2a

Here, a = 1, b = 10, and c = 41. Plugging in these values:

r = (-10 ± √(10² - 4(1)(41))) / 2(1)

r = (-10 ± √(100 - 164)) / 2

Since the discriminant (b² - 4ac) is negative, the roots will be complex:

r = (-10 ± √(-64)) / 2

r = -5 ± 4i

Now that we have the complex roots, we can write the general solution as:

y(x) = c1 * e^(-5x) * cos(4x) + c2 * e^(-5x) * sin(4x)

Here, c1 and c2 are arbitrary constants, and x is the independent variable.

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The correct pair of constraints to add is option d: P24 - P25 <= 80; P25 - P24 <= 80

To specify that production of product 2 in period 4 and in period 5 differs by no more than 80 units, the correct pair of constraints to add is option d: P24 - P25 <= 80; P25 - P24 <= 80.

The constraint P24 - P25 <= 80 ensures that the production of product 2 in period 4 (P24) does not exceed the production in period 5 (P25) by more than 80 units.

The constraint P25 - P24 <= 80 ensures that the production in period 5 (P25) does not exceed the production in period 4 (P24) by more than 80 units.

These two constraints together ensure that the production of product 2 in period 4 and period 5 differs by no more than 80 units in either direction, as both P24 - P25 and P25 - P24 are limited to be less than or equal to 80.

Therefore, the correct pair of constraints to add is option d: P24 - P25 <= 80; P25 - P24 <= 80

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Therefore, the area of the region inside the circle and outside the cardioid is. [tex]2\sqrt(3)[/tex].

To find the area of the region inside the circle and outside the cardioid, we need to integrate the difference between the areas of the circle and the cardioid over the interval where they intersect. The points of intersection are at theta = pi/6 and theta = 5pi/6, as given in the problem.

First, let's find the equation of the cardioid in Cartesian coordinates. We have r = 3 + 3sin(θ), so in Cartesian coordinates, this is:

[tex]x^2 + y^2[/tex]= [tex](3 + 3sin(θ)) ^2[/tex]

[tex]x^2 + y^2[/tex]= [tex]9 + 18sin(θ) + 9sin^2(θ)[/tex]

[tex](x^2 + y^2 - 9)[/tex] = [tex]18sin(θ) + 9sin^2(θ)[/tex]

Using the equation of the circle, r = 9sin(theta), we can rewrite sin(theta) as r/9:

([tex]x^2 + y^2 - 9) = 18(r/9) + 9(r/9)^2[/tex]

[tex]x^2 + y^2 = 3r + r^2/3[/tex]

Now we can set up the integral to find the area:

A = 1/2 ∫[tex](pi/6) ^{(5\pi/6)} [81sin^2(θ) - 9 - 18sin(θ) - 9sin^2(θ)] dθ[/tex]

[tex]A = 1/2 ∫(pi/6)^(5pi/6) [72sin^2(θ) - 9 - 18sin(θ)] dθ[/tex]

Since the region is symmetric about the vertical axis theta = pi/2, we can double this integral:

A = ∫[tex](pi/6)^(pi/2) [72sin^2(θ) - 9 - 18sin(θ)] dθ[/tex]

Now we can use the identity sin^2(θ) = 1/2(1 - cos(2θ)) to simplify the integral:

A = ∫[tex](\pi/6) ^(pi/2) [36(1-cos(2θ)) - 9 - 18sin(θ)] dθ[/tex]

A = ∫[tex](pi/6) ^(\pi/2) [-36cos(2θ) - sin(θ)] dθ[/tex]

Integrating, we get:

A = [-[tex]18sin(2θ) - cos(θ)] |_\pi/6^\pi/2[/tex]

[tex]A = [-18sin(2(\pi/2) - 2(\pi/6)) - cos(\pi/2) + cos(\pi/6)] - [-18sin(2(\pi/6)) - cos(\pi/6)][/tex]

[tex]A = [-18sin(\pi /3) - 0.5] - [-9\sqrt(3)/2 - sqrt(3)/2][/tex]

[tex]A = -18\sqrt(3)/2 + 4.5 + 9\sqrt(3)/2 - \sqrt(3)/2[/tex]

[tex]A = 4\sqrt(3)/2[/tex]

[tex]A = 2\sqrt(3)[/tex]

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The value of a₃ is equal to 8.

To find a₃ using the given terms, aₓ = a₂x-1 and a₁ = 2, follow these steps:

Step 1: Identify the value of x when finding a₃.
Since you want to find a₃, the value of x will be 3.

Step 2: Use the given formula to find a₃.
The formula provided is aₓ = a₂x-1.

Plug in the value of x as 3:

a₃ = a₂(3)-1.

Step 3: Simplify the formula.
Simplify the formula as follows:

a₃ = a₁(4).

Step 4: Substitute the given value of a₁ into the formula.
You're given that a₁ = 2, so substitute it into the simplified formula:

a₃ = 2(4).

Step 5: Solve for a₃.
To find a₃, multiply the values:

a₃ = 8.

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